Analytic derivatives and the computation of GARCH estimates

In the context of univariate GARCH models we show how analytic first and second derivatives of the log-likelihood can be successfully employed for estimation purposes. Maximum likelihood GARCH estimation usually relies on the numerical approximation to the log-likelihood derivatives, on the grounds that an exact analytic differentiation is much too burdensome. We argue that this is not the case and that the computational benefit of using the analytic derivatives (first and second) may be substantial. Furthermore, we make a comparison of various gradient algorithms that are used for the maximization of the GARCH Gaussian likelihood. We suggest the implementation of a globally efficient computation algorithm that is obtained by suitably combining the use of the estimated information matrix with that of the exact Hessian during the maximization process. As this would appear a straightforward extension, we then study the finite sample performance of the exact Hessian and its approximations (that is, the estimated information, outer products and misspecification robust matrices) in inference.     

BAYESIAN INFERENCE METHODS FOR UNIVARIATE AND MULTIVARIATE GARCH MODELS: A SURVEY

This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non‐parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe better the uncertainty about future volatilities. Finally, the survey presents an illustration using real data to show the flexibility and usefulness of the non‐parametric approach.     

A conditionally heteroskedastic independent factor model with an application to financial stock returns

We propose a new conditionally heteroskedastic factor model, the GICA-GARCH model, which combines independent component analysis (ICA) and multivariate GARCH (MGARCH) models. This model assumes that the data are generated by a set of underlying independent components (ICs) that capture the co-movements among the observations, which are assumed to be conditionally heteroskedastic. The GICA-GARCH model separates the estimation of the ICs from their fitting with a univariate ARMA-GARCH model. Here, we will use two ICA approaches to find the ICs: the first estimates the components, maximizing their non-Gaussianity, while the second exploits the temporal structure of the data. After estimating and identifying the common ICs, we fit a univariate GARCH model to each of them in order to estimate their univariate conditional variances. The GICA-GARCH model then provides a new framework for modelling the multivariate conditional heteroskedasticity in which we can explain and forecast the conditional covariances of the observations by modelling the univariate conditional variances of a few common ICs. We report some simulation experiments to show the ability of ICA to discover leading factors in a multivariate vector of financial data. Finally, we present an empirical application to the Madrid stock market, where we evaluate the forecasting performances of the GICA-GARCH and two additional factor GARCH models: the orthogonal GARCH and the conditionally uncorrelated components GARCH.     

Copula-based multivariate GARCH model with uncorrelated dependent errors

Multivariate GARCH (MGARCH) models are usually estimated under multivariate normality. In this paper, for non-elliptically distributed financial returns, we propose copula-based multivariate GARCH (C-MGARCH) model with uncorrelated dependent errors, which are generated through a linear combination of dependent random variables. The dependence structure is controlled by a copula function. Our new C-MGARCH model nests a conventional MGARCH model as a special case. The aim of this paper is to model MGARCH for non-normal multivariate distributions using copulas. We model the conditional correlation (by MGARCH) and the remaining dependence (by a copula) separately and simultaneously. We apply this idea to three MGARCH models, namely, the dynamic conditional correlation (DCC) model of Engle [Engle, R.F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350], the varying correlation (VC) model of Tse and Tsui [Tse, Y.K., Tsui, A.K., 2002. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. Journal of Business and Economic Statistics 20, 351–362], and the BEKK model of Engle and Kroner [Engle, R.F., Kroner, K.F., 1995. Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150]. Empirical analysis with three foreign exchange rates indicates that the C-MGARCH models outperform DCC, VC, and BEKK in terms of in-sample model selection and out-of-sample multivariate density forecast, and in terms of these criteria the choice of copula functions is more important than the choice of the volatility models.     

Static and dynamic models for multivariate distribution forecasts: Proper scoring rule tests of factor-quantile versus multivariate GARCH models

Many static and dynamic models exist to forecast Value-at-Risk and other quantile-related metrics used in financial risk management. Industry practice favours simpler, static models such as historical simulation or its variants. Most academic research focuses on dynamic models in the GARCH family. While numerous studies examine the accuracy of multivariate models for forecasting risk metrics, there is little research on accurately predicting the entire multivariate distribution. However, this is an essential element of asset pricing or portfolio optimization problems having non-analytic solutions. We approach this highly complex problem using various proper multivariate scoring rules to evaluate forecasts of eight-dimensional multivariate distributions: exchange rates, interest rates and commodity futures. This way, we test the performance of static models, namely, empirical distribution functions and a new factor-quantile model with commonly used dynamic models in the asymmetric multivariate GARCH class.     

Thresholds,news impact surfaces and dynamic asymmetric multivariate GARCH

Dynamic Asymmetric Multivariate GARCH (DAMGARCH) is a new model that extends the Vector ARMA‐GARCH (VARMA‐GARCH) model of Ling and Mc Aleer (2003) by introducing multiple thresholds and time‐dependent structure in the asymmetry of the conditional variances. Analytical expressions for the news impact surface implied by the new model are also presented. DAMGARCH models the shocks affecting the conditional variances on the basis of an underlying multivariate distribution. It is possible to model explicitly asset‐specific shocks and common innovations by partitioning the multivariate density support. This article presents the model structure, describes the implementation issues, and provides the conditions for the existence of a unique stationary solution, and for consistency and asymptotic normality of the quasi‐maximum likelihood estimators. The article also presents an empirical example to highlight the usefulness of the new model.     

Temporal aggregation of multivariate GARCH processes

This paper derives results for the temporal aggregation of multivariate GARCH(1,1) processes in the general vector specification. It is shown that the class of weak multivariate GARCH(1,1) processes is closed under temporal aggregation. Fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables. In some aspects, the aggregation characteristics of multivariate GARCH processes are shown to be different from those of vector autoregressive moving average processes. A numerical example illustrates some of the results.     

Bayesian semiparametric multivariate GARCH modeling

This paper proposes a Bayesian nonparametric modeling approach for the return distribution in multivariate GARCH models. In contrast to the parametric literature the return distribution can display general forms of asymmetry and thick tails. An infinite mixture of multivariate normals is given a flexible Dirichlet process prior. The GARCH functional form enters into each of the components of this mixture. We discuss conjugate methods that allow for scale mixtures and nonconjugate methods which provide mixing over both the location and scale of the normal components. MCMC methods are introduced for posterior simulation and computation of the predictive density. Bayes factors and density forecasts with comparisons to GARCH models with Student-t$t$ innovations demonstrate the gains from our flexible modeling approach.     

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA–GARCH are summarized. Various new ARCH–type models, including double threshold ARCH and GARCH, ARFIMA–GARCH, CHARMA and vector ARMA–GARCH, are also reviewed.     

The common and specific components of dynamic volatility

This paper develops a dynamic approximate factor model in which returns are time-series heteroskedastic. The heteroskedasticity has three components: a factor-related component, a common asset-specific component, and a purely asset-specific component. We develop a new multivariate GARCH model for the factor-related component. We develop a univariate stochastic volatility model linked to a cross-sectional series of individual GARCH models for the common asset-specific component and the purely asset-specific component. We apply the analysis to monthly US equity returns for the period January 1926 to December 2000. We find that all three components contribute to the heteroskedasticity of individual equity returns. Factor volatility and the common component in asset-specific volatility have long-term secular trends as well as short-term autocorrelation. Factor volatility has correlation with interest rates and the business cycle.     

The excess co-movement of commodity prices reconsidered

This paper provides an empirical reconsideration of evidence for excess co-movement of commodity prices within the framework of univariate and multivariate GARCH(1, 1) models. Alternative formulations of zero excess co-movement are provided, and corresponding score and likelihood ratio tests are developed. Monthly time series data for two sample periods, 1960–85 and 1974–92, on up to nine commodities are used. In contrast to earlier work, only weak evidence of excess co-movement is found.     

The emperor's new clothes: a critique of the multivariate t regression model

Zellner (1976) proposed a regression model in which the data vector (or the error vector) is represented as a realization from the multivariate Student t distribution. This model has attracted considerable attention because it seems to broaden the usual Gaussian assumption to allow for heavier-tailed error distributions. A number of results in the literature indicate that the standard inference procedures for the Gaussian model remain appropriate under the broader distributional assumption, leading to claims of robustness of the standard methods. We show that, although mathematically the two models are different, for purposes of statistical inference they are indistinguishable. The empirical implications of the multivariate t model are precisely the same as those of the Gaussian model. Hence the suggestion of a broader distributional representation of the data is spurious, and the claims of robustness are misleading. These conclusions are reached from both frequentist and Bayesian perspectives.     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

Forecasting crude oil price volatility

We use high-frequency intra-day realized volatility data to evaluate the relative forecasting performances of various models that are used commonly for forecasting the volatility of crude oil daily spot returns at multiple horizons. These models include the RiskMetrics, GARCH, asymmetric GARCH, fractional integrated GARCH and Markov switching GARCH models. We begin by implementing Carrasco, Hu, and Ploberger’s (2014) test for regime switching in the mean and variance of the GARCH(1, 1), and find overwhelming support for regime switching. We then perform a comprehensive out-of-sample forecasting performance evaluation using a battery of tests. We find that, under the MSE and QLIKE loss functions: (i) models with a Student’s t innovation are favored over those with a normal innovation; (ii) RiskMetrics and GARCH(1, 1) have good predictive accuracies at short forecast horizons, whereas EGARCH(1, 1) yields the most accurate forecasts at medium horizons; and (iii) the Markov switching GARCH shows a superior predictive accuracy at long horizons. These results are established by computing the equal predictive ability test of Diebold and Mariano (1995) and West (1996) and the model confidence set of Hansen, Lunde, and Nason (2011) over the entire evaluation sample. In addition, a comparison of the MSPE ratios computed using a rolling window suggests that the Markov switching GARCH model is better at predicting the volatility during periods of turmoil.     

DCC- and DECO-HEAVY: Multivariate GARCH models based on realized variances and correlations

This paper introduces the scalar DCC-HEAVY and DECO-HEAVY models for conditional variances and correlations of daily returns based on measures of realized variances and correlations built from intraday data. Formulas for multi-step forecasts of conditional variances and correlations are provided. Asymmetric versions of the models are developed. An empirical study shows that in terms of forecasts the scalar HEAVY models outperform the scalar BEKK-HEAVY model based on realized covariances and the scalar BEKK, DCC, and DECO multivariate GARCH models based exclusively on daily data.     

Estimating and predicting multivariate volatility thresholds in global stock markets

We propose a general double tree structured AR‐GARCH model for the analysis of global equity index returns. The model extends previous approaches by incorporating (i) several multivariate thresholds in conditional means and volatilities of index returns and (ii) a richer specification for the impact of lagged foreign (US) index returns in each threshold. We evaluate the out‐of‐sample forecasting power of our model for eight major equity indices in comparison to some existing volatility models in the literature. We find strong evidence for more than one multivariate threshold (more than two regimes) in conditional means and variances of global equity index returns. Such multivariate thresholds are affected by foreign (US) lagged index returns and yield a higher out‐of‐sample predictive power for our tree structured model setting. Copyright © 2006 John Wiley & Sons, Ltd.     

Generalized spectral testing for multivariate continuous-time models

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples.     

Estimating nonlinear time-series models using simulated vector autoregressions

This paper develops two new methods for conducting formal statistical inference in nonlinear dynamic economic models. The two methods require very little analytical tractability, relying instead on numerical simulation of the model's dynamic behaviour. Although one of the estimators is asymptotically more efficient than the other, a Monte Carlo study shows that, for a specific application, the less efficient estimator has smaller mean squared error in samples of the size typically encountered in macroeconomics. The estimator with superior small sample performance is used to estimate the parameters of a real business cycle model using observed US time-series data.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Testing multivariate distributions in GARCH models

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation eliminates the effect of parameter estimation, allowing a distribution-free test statistic to be constructed. We show that the K-transformation takes a very simple form for testing multivariate normal and multivariate t-distributions. The procedure is applied to a multivariate financial time series data set.